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Câu hỏi: Cho nguyên hàm $I=\int{\ln \text{xdx}}$. Tìm đẳng thức đúng?
A. $I=\ln x+\int{d\text{x}}$
B. $I=x\ln \text{x}+\int{d\text{x}}$
C. $I=\ln \text{x}-\int{d\text{x}}$
D. $I=x\ln \text{x}-\int{d\text{x}}$
A. $I=\ln x+\int{d\text{x}}$
B. $I=x\ln \text{x}+\int{d\text{x}}$
C. $I=\ln \text{x}-\int{d\text{x}}$
D. $I=x\ln \text{x}-\int{d\text{x}}$
Ta có: $I=\int{\ln \text{xdx}}$.
Đặt: $\left\{ \begin{aligned}
& u=\ln \text{x} \\
& dv=d\text{x} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=\dfrac{d\text{x}}{x} \\
& v=x \\
\end{aligned} \right.$
Khi đó: $I=uv-\int{v\text{d}u}=x\ln \text{x}-\int{d\text{x}}$.
Đặt: $\left\{ \begin{aligned}
& u=\ln \text{x} \\
& dv=d\text{x} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=\dfrac{d\text{x}}{x} \\
& v=x \\
\end{aligned} \right.$
Khi đó: $I=uv-\int{v\text{d}u}=x\ln \text{x}-\int{d\text{x}}$.
Đáp án D.